This section describes the methods for determining the links between the graphical properties of the linked chemical graph and the thermodynamic parameters of Iron Telluride \(({\rm FeTe}_2)\). Then, the HoF of Iron Telluride \(({\rm FeTe}_2)\) is calculated for various formula unit cells of Iron Telluride. The change in “enthalpy of formation,” or (HoF) is the transformation that occurs when a mole of a molecule is broken down into its component parts in their natural state28. This transformation usually occurs at a specific temperature and pressure (usually \(25C + 1\) atmosphere). The notation ‘\(\delta H_f\)’ is commonly used to refer to the HoF. The HoF is expressed in terms of the energy per mole (inkilojoules/mol) or kilocalories/mol(inkcal/mol) of \({\rm FeTe}_2\). Divide \({\rm FeTe}_2 = -51.9/ -65.8 kJ/mol\) by Avogadro’s number, \(6.02214\times 10^{23}\, \textrm{mol}^{-1}\).Topological indices are shown graphically for different formula unit cells. When fitting rational graphical models, the output variable is enthalpy, and the input variables are topological co-indices. Last but not least, many of the curves are fitted with the help of the curve fitting tool in MATLAB. Most of our built-in curve fitting techniques are used on our data. Let’s say we have a data collection and we have a number of observations of (n). Let’s also let’s say g is a set of all the fitted values that match Y. Let’s also think about the standard deviation. This is a key component that tells us how far our values differ from the mean. We can get a more precise fit with the standard deviation. To compute an error, let’s use the standard deviation. It can be expressed as a square root of the error value. Here, \(\backslash RMSE\)” stands for “standard deviation of residuals.” This test tells us how far the residuals differ from the model’s predicted values. It tells us how far away the residuals are from the mean. It’s easy to read this test as a mean squared error because it’s just Total squared error An extra statistical test is (SSE). The degree to which observed values differ from our fitted curve is examined using the \(R^2\)-test. A good match is shown by \(R^2\) nearing 1, whereas a poor estimate is indicated by \(R^2\) approaching 0. The ratio of estimated variance to actual variance is denoted by \(R^2\). We will only investigate these three statistical tests, despite the fact that there are a few more in the literature, because MATLAB’s Rational Curve Fitting tool selects the model based on and only29.Below are the models that we have discussed for indices and correlation HOF. We have also used MATLAB to display the graphic representations as well as the curve that fits the statistical parameters. Figures 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 and 19. We constructed rational curves using MATLAB’s curve fitting toolkit.models using Tables 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17. MATLAB tools were used to produce key performance measures, such as \(\mathbb {RMSE}\), \(\mathbb {SSE}\), \(\mathfrak {r^2}\), and \(adj\mathfrak {(r^2)}\), to evaluate the explanatory power and accuracy of the fitted models.$$\begin{aligned} f(\Re {R_1}) =\frac{(\chi _1\times \Re {R_1}^3 + \chi _2\times \Re {R_1}^2 + \chi _3\times \Re {R_1} + \chi _4)}{(\Re {R_1}^3 + \Psi _1\times \Re {R_1}^2 + \Psi _2\times \Re {R_1} + \Psi _3)} \end{aligned}$$In which the Normalized mean of \(\Re {R_1}\) is 222.5 and the standard deviation is 171.6. The coefficients are: \(\kappa _1= -2.431\), with \({\mathbb {C}}_{b}=(-4.717, -0.1442)\), \(\Psi _1= 1.233\) with \({\mathbb {W}}_{b}=(0.5785, 1.889)\), \(\Psi _2= 0.196\) with \({\mathbb {W}}_{b}=(-0.2597, 0.6518)\), \(\chi _2= -2.356\), with \({\mathbb {W}}_{b}=(-6.567, 1.854)\), \(\kappa _3= 0.2201\) with \({\mathbb {W}}_{b}=(-1.498, 1.938)\), \(\kappa _4= 0.1682\),with \({\mathbb {W}}_{b}=(-1.803, 2.14)\), \(\Psi _3= -0.026\) with \({\mathbb {W}}_{b}=(-0.2623, 0.2103)\).Fig. 8HOF of \(\Re {R_{1}}(FeTe_{2})\).Table 6 HOF and \(\Re {R_{1}}(FeTe_{2})\).$$\begin{aligned} f(\Re {R_{-1}}) =\frac{(\kappa _1\times \Re {R_{-1}}^3 + \kappa _2\times \Re {R_{-1}}^2 + \kappa _3\times \Re {R_{-1}} + \kappa _4)}{(\Re {R_{-1}}^3 + \Psi _1\times \Re {R_{-1}}^2 + \Psi _2\times \Re {R_{-1}} + \Psi _3)} \end{aligned}$$In which Normalized mean of \(\Re {R_{-1}}\) is 135.4 and standard deviation is 125.6. The coefficients are: \(\kappa _1= -2.385\), with \({\mathbb {W}}_{b}=(-4.822, 0.05173)\), \(\kappa _2= -2.581\), with \({\mathbb {W}}_{b}=(-6.816, 1.655)\), \(\Psi _1= 1.33\) with \({\mathbb {W}}_{b}=(0.7634, 1.896)\), \(\kappa _3= -0.1022\), with \({\mathbb {W}}_{b}=(-1.848, 1.643)\),\(\kappa _4= 0.1776\), with \({\mathbb {W}}_{b}=(-1.616, 1.971)\), \(\Psi _2= 0.34\) with \({\mathbb {W}}_{b}=(-0.07087, 0.7509)\), \(\Psi _3= -0.008958\) with \({\mathbb {W}}_{b}=(-0.2082, 0.1903)\).Table 7 CF between HOF and \(\Re {R_{-1}}(FeTe_{2})\).Fig. 9HOF of \(\Re {R_{-1}(FeTe_{2})}\).$$\begin{aligned} f(\Re {R_\frac{1}{2}}) =\frac{(\kappa _1\times \Re {R_\frac{1}{2}}^3 + \kappa _2\times \Re {R_\frac{1}{2}}^2 + \kappa _3\times \Re {R_\frac{1}{2}} + \kappa _4)}{(\Re {R_\frac{1}{2}}^3 + \Psi _1\times \Re {R_\frac{1}{2}}^2 + \Psi _2\times \Re {R_\frac{1}{2}} + \Psi _3)} \end{aligned}$$In which normalized mean of \(\Re {R_\frac{1}{2}}\) is 181 and standard deviation is 150.4. The coefficients are: \(\kappa _1= -2.413\), with \({\mathbb {W}}_{b}=(-4.752, -0.07324)\), \(\kappa _2= -2.442\), with \(\Psi _1= 1.27\) with \({\mathbb {W}}_{b}=(0.6477, 1.893)\), \(\Psi _2= 0.2513\) with \({\mathbb {W}}_{b}=(-0.1896, 0.6922)\), \({\mathbb {W}}_{b}=(-6.663, 1.779)\), \(\kappa _3= 0.09659\) with \({\mathbb {W}}_{b}=(-1.63, 1.823)\), \(\kappa _4= 0.1742\) with \({\mathbb {W}}_{b}=(-1.728, 2.076)\), \(\Psi _3= -0.0204\) with \({\mathbb {W}}_{b}=(-0.2414, 0.2006)\).Fig. 10HOF of \(\Re {R_\frac{1}{2}(FeTe_{2})}\).Table 8 CF between HOF and \(\Re {R_\frac{1}{2}(FeTe_{2})}\).$$\begin{aligned} f(\Re {R_\frac{-1}{2}}) = \frac{(\kappa _1\times \Re {R_\frac{-1}{2}}^3 + \kappa _2\times \Re {R_\frac{-1}{2}}^2 + \kappa _3\times \Re {R_\frac{-1}{2}} + \kappa _4)}{(\Re {R_\frac{-1}{2}}^3 + \Psi _1\times \Re {R_\frac{-1}{2}}^2 + \Psi _2\times \Re {R_\frac{-1}{2}} + \Psi _3)} \end{aligned}$$Fig. 11HOF of \(\Re {R_\frac{-1}{2}}(FeTe_{2})\).In which Normalized mean of \(\Re {R_\frac{-1}{2}}\) is 143.8 and standard deviation is 130.3. The coefficients are: \(\kappa _1= -2.391\), with \({\mathbb {W}}_{b}=(-4.805, 0.02294)\), \(\kappa _2= -2.55\) with \(\Psi _1= 1.317\) with \({\mathbb {W}}_{b}=(0.7374, 1.896)\), \({\mathbb {W}}_{b}=(-6.783, 1.683)\), \(\kappa _3= -0.05842\), with \({\mathbb {W}}_{b}=(-1.799, 1.682)\), \(\kappa _4= 0.1775\), with \({\mathbb {W}}_{b}=(-1.64, 1.995)\), \(\Psi _2= 0.3205\) with \({\mathbb {W}}_{b}=(-0.09767, 0.7386)\), \(\Psi _3= -0.01173\) with \({\mathbb {W}}_{b}=(-0.2154, 0.192)\).Table 9 HOF and \(\Re {R_\frac{-1}{2}}(FeTe_{2})\).Fig. 12HOF of \(\Re {M_1}(FeTe_{2})\).$$\begin{aligned} f(\Re {M_1}) = \frac{(\kappa _1\times \Re {M_1}^3 + \kappa _2\times \Re {M_1}^2 + \kappa _3\times \Re {M_1} + \kappa _4)}{(\Re {M_1}^3 + \Psi _1\times \Re {M_1}^2 + \Psi _2\times \Re {M_1} + \Psi _3)} \end{aligned}$$In which Normalized mean of \(\Re {M_1}\) is 369 and standard deviation is 304.6. The coefficients are: \(\kappa _1= -2.415\), with \({\mathbb {W}}_{b}=(-4.749, -0.08057)\), \(\kappa _2= -2.434\), with \({\mathbb {W}}_{b}=(-6.653, 1.786)\), \(\Psi _1= 1.267\) with \({\mathbb {W}}_{b}=(0.6407, 1.892)\), \(\kappa _3= 0.109\) with \({\mathbb {W}}_{b}=(-1.617, 1.835)\), \(\kappa _4= 0.1737\), with \({\mathbb {W}}_{b}=(-1.735, 2.083)\), \(\Psi _2= 0.2458\) with \({\mathbb {W}}_{b}=(-0.1968, 0.6883)\), \(\Psi _3= -0.02102\) with \({\mathbb {W}}_{b}=(-0.2435, 0.2014)\).Table 10 HOF and \(\Re {M_1}(FeTe_{2})\).$$\begin{aligned} f(\Re {M_2}) = \frac{(\kappa _1\times \Re {M_2}^3 + \kappa _2\times \Re {M_2}^2 + \kappa _3\times \Re {M_2} + \kappa _4)}{(\Re {M_2}^3 + \Psi _1\times \Re {M_2}^2 + \Psi _2\times \Re {M_2} + \Psi _3)} \end{aligned}$$In which the Normalized mean of \(\Re {M_2}\) is 222.5 and the standard deviation is 171.6. The coefficients are: \(\kappa _1= -2.431\), with \({\mathbb {W}}_{b}=(-4.717, –0.1442)\), \(\kappa _2= -2.356\), \(\Psi _1= 1.233\) with \({\mathbb {W}}_{b}=(-6.567, 1.854)\), \(\kappa _3= 0.2201\) with \({\mathbb {W}}_{b}=(-1.498, 1.938)\), \(\kappa _4= 0.1682\) with \({\mathbb {W}}_{b}=(-1.803, 2.14)\), with \({\mathbb {W}}_{b}=(0.5785, 1.888)\), \(\Psi _2= 0.196\) with \({\mathbb {W}}_{b}=(-0.2597, 0.6518)\), \(\Psi _3= -0.026\) with \({\mathbb {W}}_{b}=(-0.2623, 0.2103)\).Fig. 13HOF of \(\Re {M_2}(FeTe_{2})\).Table 11 CF between HOF and \(\Re {M_2}(FeTe_{2})\).$$\begin{aligned} f(\Re {HM}) =\frac{(\kappa _1\times \Re {HM}^3 + \kappa _2\times \Re {HM}^2 + \kappa _3\times \Re {HM} + \kappa _4)}{(\Re {HM}^3 + \Psi _1\Re {HM}^2+\Psi _2\Re {HM}+\Psi _3)} \end{aligned}$$In which the normalized mean of \(\Re {HM}\) is 937 and the standard deviation is 710.4. The coefficients are: \(\kappa _1= -2.435\), with \({\mathbb {W}}_{b}=(-4.709, -0.1606)\), \(\Psi _1= 1.224\) with \({\mathbb {W}}_{b}=(0.562, 1.887)\), \(\Psi _2= 0.1827\) with \({\mathbb {W}}_{b}=(-0.2763, 0.6416)\),\(\kappa _2= -2.336\), with \({\mathbb {W}}_{b}=(-6.543, 1.872)\), \(\kappa _3= 0.2499\), with \({\mathbb {W}}_{b}=(-1.466, 1.966)\), \(\kappa _4= 0.1663\), with \({\mathbb {W}}_{b}=(-1.822, 2.155)\), \(\Psi _3= -0.02718\) with \({\mathbb {W}}_{b}=(-0.2674, 0.2131)\).Fig. 14HOF of \(\Re {HM}(FeTe_{2})\).Table 12 CF between HOF and \(\Re {HM}(FeTe_{2})\).$$\begin{aligned} f(\Re {ABC}) =\frac{(\kappa _1\times \Re {ABC}^5 + \kappa _2\times \Re {ABC}^4 + \kappa _3\times \Re {ABC}^3 + \kappa _4\times \Re {ABC}^2 + \kappa _5\times \Re {ABC} + \kappa _6)}{(\Re {ABC} + \Psi _1)} \end{aligned}$$Fig. 15HOF of \(\Re {ABC}(FeTe_{2})\).In which the normalized mean of \(\Re {ABC}\) is 27.12 and the standard deviation is 15.86. The coefficients are: \(\kappa _1= -2.051\), with \({\mathbb {W}}_{b}=(-2.489, -1.614)\), \(\kappa _2= 1.785\) with \({\mathbb {W}}_{b}=(1.312, 2.258)\), \(\kappa _5= -4.27\) with \({\mathbb {W}}_{b}=(-4.736, -3.805)\), \(\kappa _6= 2.654\) with \({\mathbb {W}}_{b}=(2.254, 3.054)\), \(\kappa _=3 4.811\) with \({\mathbb {W}}_{b}=(3.783, 5.839)\), \(\kappa _4= -3.963\) with \({\mathbb {W}}_{b}=(-4.964, -2.962)\), \(\Psi _1= -0.3968\), with \({\mathbb {W}}_{b}=(-0.4378, -0.3559)\).Table 13 CF between HOF and \(\Re {ABC}(FeTe_{2})\).$$\begin{aligned} f(\Re {GA}) =\frac{(\kappa _1\times \Re {GA}^3 + \kappa _2\times \Re {GA}^2 + \kappa _3\times \Re {GA} + \kappa _4)}{(\Re {GA}^3 + \Psi _1\times \Re {GA}^2 + \Psi _2\times \Re {GA} + \Psi _3)} \end{aligned}$$In which the normalized mean of \(\Re {GA}\) is 155.4 and the standard deviation is 136.7. The coefficients are: \(\kappa _1= -2.399\), with \({\mathbb {W}}_{b}= (-4.785, -0.01203)\), \(\kappa _2= -2.512\), with \({\mathbb {W}}_{b}=(-6.741, 1.1717)\), \(\Psi _1= 1.3\) with \({\mathbb {W}}_{b}=(0.7053, 1.895)\), \(\Psi _2= 0.2961\), with \({\mathbb {W}}_{b}=(-0.1307, 0.7228)\),\(\kappa _3= -0.00363\), with \({\mathbb {W}}_{b}=(-1.739, 1.731)\), \(\kappa _4= 0.1768\), with \({\mathbb {W}}_{b}=(-1.67, 2.024)\), \(\Psi _3= -0.015\) with \({\mathbb {W}}_{b}=(-0.015, 0.1945)\).Table 14 CF between HOF and \(\Re {GA}(FeTe_{2})\).Fig. 16HOF of \(\Re {GA}(FeTe_{2})\).$$\begin{aligned} f(\Re {PM_1}) = \frac{(\kappa _1\times \Re {PM_1}^4 + \kappa _2\times \Re {PM_1}^3 + \kappa _3\times \Re {PM_1}^2 + \kappa _4\times \Re {PM_1} + \kappa _5)}{(\Re {PM_1} + \Psi _1)} \end{aligned}$$In which the normalized mean of \(\Re {PM_1}\) is \(4.446e+08\) and the standard deviation is \(6.376e+08\). The coefficients are: \(\kappa _1= 36.73\), with \({\mathbb {W}}_{b}=(-572.5, 646)\), \(\kappa _2= -65.91\), with \({\mathbb {W}}_{b}=(-1156, 1024)\), , \(\kappa _5= 1.853\), \(\kappa _3= -31.2\) with \({\mathbb {W}}_{b}=(-551.4, 489)\), \(\kappa _4= 27.52\) with \({\mathbb {W}}_{b}=(-457.4, 512.4)\) with \({\mathbb {W}}_{b}=(-44.13, 47.84)\), \(\Psi _1= 1.565\) with \({\mathbb {W}}_{b}=(-16.02, 19.15)\).Fig. 17HOF of \(\Re {PM_1}(FeTe_{2})\).Table 15 HOF and \(\Re {PM_1}(FeTe_{2})\).$$\begin{aligned} f(\Re {PM_2}) = \frac{(\kappa _1\times \Re {PM_2}^4 + \kappa _2\times \Re {PM_2}^3 + \kappa _3\times \Re {PM_2}^2 + \kappa _4\times \Re {PM_2} + \kappa _5)}{(\Re {PM_2} + \Psi _1)} \end{aligned}$$In which the normalized mean of \(\Re {PM_2}\) is \(1.334e+08\) and the standard deviation is \(1.913e+08\). The coefficients are: \(\kappa _1= 36.73\), with \({\mathbb {W}}_{b}=(-572.5, 645.9)\), \(\kappa _2= -65.91\), with \({\mathbb {W}}_{b}=(-1156, 1024)\), \(\kappa _3= -31.2\) with \({\mathbb {W}}_{b}=(-551.4, 489)\), \(\kappa _4= 27.52\) with \({\mathbb {W}}_{b}=(-457.4, 512.4)\), \(\kappa _5= 1.853\) with \({\mathbb {W}}_{b}=(-44.13, 47.84)\), \(\Psi _1= 1.565\) with \({\mathbb {W}}_{b}= (-16.02, 19.15)\).Fig. 18HOF of \(\Re {PM_2}(FeTe_{2})\).Table 16 CF between HOF and \(\Re {PM_2}(FeTe_{2})\).$$\begin{aligned} f(\Re {F}) =\frac{(\kappa _1\times \Re {F}^3 +\kappa _2\Re {F}^2+\kappa _3\times \Re {F}+\kappa _4 )}{(\Re {F}^3+\Psi _1\Re {F}^2 + \Psi _2\Re {F} + \Psi _3)} \end{aligned}$$In which the normalized mean of \(\Re {F}\) is 492 and the standard deviation is 367.3. The coefficients are: \(\kappa _1= -2.439\), with \({\mathbb {W}}_{b}=(-4.702, -0.1757)\), \(\Psi _1= 1.216\) with \({\mathbb {W}}_{b}=(0.5467, 1.885)\), \(\kappa _2= -2.316\), with \({\mathbb {W}}_{b}=(-6.522, 1.889)\), \(\kappa _3= 0.2778\) with \({\mathbb {W}}_{b}=(-1.437, 1.992)\), \(\kappa _4= 0.1644\) with \({\mathbb {W}}_{b}=(-1.84, 2.169)\), \(\Psi _2= 0.1702\) with \({\mathbb {W}}_{b}=(-0.2916, 0.6319)\), \(\Psi _3= -0.02821\) with \({\mathbb {W}}_{b}=(-0.2722, 0.2158)\).Table 17 CF between HOF and \(\Re {F}(FeTe_{2})\).Fig. 19HOF of \(\Re {F}(FeTe_{2})\).
